Supply Chain Networking Models Under Fuzzy Uncertainty

— In decision-making model, the techniques of numerical analysis have been widely adopted. It is rare for someone to solve a linear program by hand — except perhaps in a class-room. Large-scale simulations would be all but impossible without the aid of a computer. For many people, numerical techniques have superseded analytic techniques as a tool for solving mathematical problems. This paper proposed Generalized LU-Exponential Trapezoidal Fuzzy Number and their ranking based on numerical integration. In this ranking method, the values are calculated with left and right spreads at some 𝜶 − level of generalized LU-exponential trapezoidal fuzzy numbers and Weddle‘s rule for numerical integration. To illustrate the proposed methods, a fuzzy four dimensional transportation problem (FDTP) is proposed and solved. This ranking approach is very simple and useful for the real life inequality based decision making problems.


Literature Review, Research gap and contributions:
The classical transportation problem (TP) was first developed by Hitchcock [19] in 1941. The solid transportation problem (STP) was put forward by Shell [20] in 1955. Chen and Chen [8] in 2007 discussed the fuzzy risk analysis on the ranking of generalized trapezoidal fuzzy numbers. Bai and Liu [26] in 2016 discussed robust optimization of supply chain network design in fuzzy decision system. Baidya et al. [23] in 2016 solved multi-stage multi-objective solid transportation problem for disaster response operation with type-2 triangular fuzzy variables. Fathian et al. [24] in 2018 proposed location and transportation planning in supply chains under uncertainty and congestion by using an improved electromagnetism-like algorithm. Khorshidian et al. [25] in 2019 developed an intelligent truck scheduling and transportation planning optimization model for product portfolio in a cross-dock. Baidya et al. [31] studied an STP with breakable items with hybrid and fuzzy safety factors using LINGO and Genetic Algorithm. Das et al. [31] in 2016 developed a breakable multi-item multi stage solid transportation problem under budget with gaussian type-2 fuzzy parameters. Sengupta et al. [31] in 2018 proposed a gamma type-2 defuzzification method for solving a solid transportation problem considering carbon emission. Roy and Midya [30] in 2019 developed multi-objective fixed-charge solid transportation problem with product blending under intuitionistic fuzzy environment. Mishra and Kumar [32] in 2020 proposed JMD method for transforming an unbalanced fully intuitionistic fuzzy transportation problem into a balanced fully intuitionistic fuzzy transportation problem. Baykasoğlu and Subulan [33] in 2019 developed a direct solution approach based on constrained fuzzy arithmetic and metaheuristic for fuzzy transportation problems. Bera and Mondal [34] in 2020 proposed Credit linked two-stage multi-objective transportation problem in rough and bi-rough environments.
This paper defines generalized LU-exponential trapezoidal fuzzy numbers and their ranking based on the approach proposed by Kumar et al. [12] in 2011 and numerical integration. The approach proposed by Kumar et al. [12] was developed with respect to ranking function : → , (where , a set of fuzzy numbers is defined on set of real numbers and ∈ 0,1 is degree of optimism, which maps each fuzzy number into the real line, where a natural order exists) to compare both normal and non-normal p-norm trapezoidal fuzzy numbers. In this proposed ranking method, the most popular quadrature formula i.e., Weddle's rule and left and right spreads at some −level of generalized LU-exponential trapezoidal fuzzy numbers are considered to find the ranking of generalized LU-exponential trapezoidal fuzzy numbers with appropriate membership function instead of definite integration used by Kumar et al. [12] to find ranking function . Looking the difficulty to handle the integration part of left and right spreads at some −level of generalized LU-exponential trapezoidal fuzzy numbers, numerical integration is taken into account in place of direct integration to calculate the ranking of generalized LU-exponential trapezoidal fuzzy numbers.

Motivation:
Roadways is known to be the most reliable modes of transport; facilitating easy and fast movement of traffic with door to door services, also having comparatively lesser construction costs than railways. Now-a-days, sources and destinations are connected by different routes. But as we break through reality we all know that not all roads are as smooth as they sound. Certain roads exist which contain potholes, cracks, or uneven surfaces that are unsafe to drive on; such roads are termed as ‗Bad Roads'. Most of the good and safe roads in India or anywhere else in the world are turned into bad roads because of being left under construction or due to natural calamities like floods, landslides or even because of simple causes like heavy rains and accidents. Even though we may ignore such roads and drive past them and need to acknowledge the fact that they are very hazardous for all human beings to transport the commodities from sources to destinations. Every day we read the paper, and we see that there are at least two to three articles about accidents caused due to the extreme conditions of roads.

Figure-4: Images of the bad roads across India
In recent decades, very few researchers Halder et al. [21] in 2017 and Aktar et al. [22] in 2020 have used this assumption and STP becomes FDTP with source, destination, conveyance and path. This has motivated to formulate and solve multi-item four dimensional solid transportation problems in both crisp and fuzzy environments.

Problems Description and Methodologies:
This paper proposed generalized LU-exponential trapezoidal fuzzy number and their ranking based on numerical integration and also two four dimensional solid transportation problems have been developed in deterministic and fuzzy environment respectively. Considering transportation paths, distances, availabilities, demands, conveyances capacities, formulate the solid transportation models under crisp and fuzzy environments to transport ℎ item through ℎ path from ℎ source to ℎ destination using ℎ conveyances. The fuzzy uncertainty has been removed by using the proposed ranking method i.e., using defuzzification technique for generalized LU-exponential trapezoidal fuzzy number. The models are illustrated numerically with LINGO.13.0 optimization software and the obtained optimal results are compared.

FUNDAMENTAL CONCEPTS
3.1. Generalized Exponential Trapezoidal Fuzzy Numbers: Generally, a generalized fuzzy number is described as any fuzzy subset of the real line R, whose membership function satisfies the following conditions, (i) is a continuous mapping from R to the closed interval [0,1], Where 0 < ≤ 1 and 1 , 2 , 3 and 4 are real numbers. We call this type of generalized fuzzy number a trapezoidal fuzzy number, and it is denoted by = 1 , 2 , 3 , 4 ; When = 1, this type of generalized fuzzy number is called normal fuzzy number and is represented by = 1 , 2 , 3 , 4 .

Numerical Integration or Quadrature
The problem of numerical integration is to evaluate = . The problem arises when the integration can't be carried out either due to the fact that is not integrable or when is known only at a finite number of points. So many methods are exists in the literature for numerical integration. Newton-Cotes quadrature formulae is one of the most popular and effective formulae for numerical integration or quadrature.

Definition-4.3: The ranking function
: → , where is a set of real fuzzy numbers and the degree of optimism ∈ 0,1 , with a natural order which maps each fuzzy number into the real number. Remark-4.1: For a pessimistic decision maker = 0, for an optimistic decision maker = 1 and for a moderate decision maker = 0.5.
Here, we can find the value of I 2 easily and which is but the value of I 1 is difficult because the integrand is a non-integrable function. To find the value of I 1 we may apply Weddle's rule (Newton-Cotes Quadrature formulae) for numerical integration or Quadrature.  ],

Example
 : available quantity of ℎ item at ℎ source (in Kg).
 : required quantity of ℎ item at ℎ destination (in Kg).
 : capacity of ℎ conveyance when going through ℎ path (in Kg).


: distance from ℎ source to ℎ destination through ℎ path (in hundred of Km).


: it is a binary auxiliary variable whose value is 1 if ℎ item is transported from ℎ source to ℎ destination by ℎ conveyance via ℎ path and 0 otherwise, i.e. = 1 > 0 0 ℎ .

Model-6.1.: Formulation of Breakable Crisp Multi-Item FDTP
Let us consider ‗M' supply points (or sources), ‗N' destination, conveyances through P number of different paths connecting each sources to each destinations. Also we consider that Q be the number of items those are transported from plants to destination by different modes of conveyances. The crisp multi-item FDTP has been formulated as follows: Objective function:

Plant capacity constraint:
The sum of the number of units transported from the plant to the distribution centers by all vehicles must be less than or equal to availability of the plant. That is, , for = 1,2, … , , = 1,2, … , 2

Model-6.3.: Formulation of Breakable Fuzzy Multi-Item FDTP
Fuzzy set theory was presented by Zadeh [1] in 1965. The theory provided a mathematical approach for dealing with imprecise concepts and problems that have many possible solutions. In real world almost everywhere we see the uncertain events and fuzzy uncertainty is a growing concept. The transportation problem is the special application of linear programming problem which aims to assign the optimal amounts of a product to be transported from various supply points to various demand points so that the total transportation cost is a minimum. Considering unit transportation cost, availabilities, demands and conveyance capacities all are fuzzy in nature, the fuzzy multi-item FDTP has been formulated as follows: Fuzzy Objective function:

Defuzzification of Model-6.3 using ranking function:
Defuzzification is the process of producing a quantifiable result in crisp logic, given fuzzy sets and corresponding membership degrees. It is the process that maps a fuzzy set to a crisp set. It is typically needed in fuzzy control systems. These will have a number of rules that transform a number of variables into a fuzzy result, that is, the result is described in terms of membership in fuzzy sets. the proposed transportation problem was Subject to the constraints Also, the deterministic form of the demand constraint (10) of the Model-6.4 is as follows:

Numerical Experiment:
A company has received a contract to supply two types of gravel for two new construction projects location in towns A and B. The company has 2 gravel pits located in towns X and Y in two different routes. Construction engineers have estimated the required amounts of gravel which will be needed at these construction projects: Projects location in town A Projects location in town B Pit located of town X B 11 =240 B 12 =132 Pit located of town Y B 21 =100 The required fuzzy amounts of gravel which will be needed at these construction projects: Projects location in town A Projects location in town B The gravel required by the construction projects can be supplied by two pits. The amount of gravel which can be supplies by each pit is as follows:

Projects location in town A Projects location in town B
Pit located of town X A 11 =447 A 12 =369 The fuzzy amount of gravel which can be supplies by each pit is as follows: Projects location in town A Projects location in town B The percentages of breakability per unit distance are given below: The distances from pits located in towns X and Y to two projects location in towns A and B in two different routes (in hundred of Km) are given in the following The conveyances capacities to transport the gravels from 2 gravel pits located in towns X and Y to two construction projects location in towns A and B with two different types of vehicles are given below: Projects location in town A Projects location in town B Problem is to schedule the shipment from each pit to each project location under crisp and fuzzy environments in such a manner so as to minimize the total transportation cost within the constraints imposed by pit capacities and project requirements.

Analysis of the results and Discussion:
When some items are produced in the firm then it's required to deliver to the customers so safely in such a way that the transportation cost has minimum. In practical, to solve fuzzy optimization problem or fuzzy decision making problem, we face some problems due to complexity of the function and in such cases numerical integration play a vital role in computational Mathematics. We know that, breakability is a part of transportation.
When breakable items are transported then due to breakability, the costing of transportation increases and to transport the breakable items need to take more care in transportation. In this paper, two such transportation problems are proposed under deterministic and fuzzy environments and as well as solved using LINGO 13.0 optimization solver (Generalized Reduced Gradient Technique). The optimal solutions of the proposed models are given below: Here, the commodities are transported from sources to destination with different mode of transport i.e., conveyances with different paths from each source to each destination so as to minimize the total transportation cost. As we know, none of the researcher have been applied the concept of numerical integration to solve the fuzzy optimization problem for fruitful optimum solution, the proposed defuzzification technique for generalized LU-exponential trapezoidal fuzzy number is highly significant and useful to solve fuzzy optimization problem like fuzzy transportation problem, fuzzy inventory problem, fuzzy traveling salesman problem etc. The four optimal solutions of the model-6.1, model-6.2, model-6.3 and model-6.4 imply that the use of numerical integration i.e., application of proposed defuzzification technique to solve fuzzy decision making problem or fuzzy optimization problem is highly fruitful and effective. As we know that breakable items are so sensitive and need to take more care in the time of transportation and for this reason, need to pay more money in

Sensitivity Analysis and Geometrical Meaning of the Model-6.2:
In this paper, the ranking method is so calculated with left and right spreads at some −level of fuzzy numbers.
In the ranking function , the degree of optimism lies between 0 and 1. For the pessimistic decision maker = 0, optimistic decision maker = 1 and moderate decision maker = 0.5. The sensitivity analysis of the Model-6.3 and Model-6.4 with respect to the degree of optimism and breakability are presented below:  In view of Transportation problems, optimism and pessimism can reflect distinguished distinction of different decision makers and also they can tend to be both compatible and enduring. It also implies that optimism and pessimism consistently influence how the decision maker responds to the decision environment. From the sensitivity analysis of the models 6.3 and 6.4 it is so much clear that the optimal cost and optimal solution are highly dependent on degree of optimism (0 ≤ ≤ 1) and since the optimal costs with respect to are fluctuating, so its decision maker's choice, what value of they would like to consider for getting the optimal solution of the proposed models. generalized LU-exponential trapezoidal fuzzy number and its ranking with left and right spreads at some −level of generalized LU-exponential trapezoidal fuzzy numbers. Since left and right spreads at some −level of generalized LU-exponential trapezoidal fuzzy numbers is difficult to calculate due to non-integrable integrand and that's why the numerical integration i.e., Weddle's rule is used to find the left and right spreads at some −level of generalized LU-exponential trapezoidal fuzzy numbers as well as to compare the two generalized LU-exponential trapezoidal fuzzy numbers. Also to validate the methodologies, the six sets of generalized LU-exponential trapezoidal fuzzy numbers are compared. The main advantage of the proposed approach is that the proposed approach provides the correct ordering of generalized LU-exponential trapezoidal fuzzy numbers and also the proposed approach is very simple and easy to apply in the real life problems. The proposed methodology to find the ranking value of the proposed fuzzy number gives some numerical error since here we developed the methodology using Weddle's rule with six numbers of sub-intervals. In future, if we can increase the number of sub-interval in the numerical integration part than we can get the more accurate ranking value of generalized LU-exponential trapezoidal fuzzy numbers. We all know that, transportation plays a vital role to grow the economy and the path of the transportation is an imperatives part like source, destination and conveyances of any transportation problem. The paths from each source to each destination are taken into account to formulate crisp as well as fuzzy transportation problem. As the distances from sources to destinations through different paths are significant in transportation problem thus, in reality, the said concept is highly acceptable. When impreciseness and complexity of the function occurs in any optimization problem at a time then it is difficult to solve the problem to obtain the fruitful optimal solution. In practical sense, the proposed defuzzification technique for generalized LU-exponential trapezoidal is more significant and useful to overcome.
The models in the paper are formulated by taking the different path of the transportation and fuzziness. Also here the fuzzy (type-1) model has been defuzzified using the proposed defuzzification technique for generalized LUexponential trapezoidal fuzzy number and in future the models and the methodologies can be developed in the following manner:

1)
Smooth transportation is possible when the routes of the transportation are safe and sound. So the proposed deterministic and fuzzy models can be formulated involving safety constraint.

2)
In the fuzzy model formulation, type-1 generalized LU-exponential trapezoidal fuzzy number has been considered but in future we can formulate the model with type-2 generalized LU-exponential trapezoidal fuzzy number.
3) Breakability and restriction on transportation is an important concept in transportation problem. So in future we can formulate the models introducing % of breakability and restriction on transportation.

4)
Global warming is a important topic in the world. Several meetings has been held among the different countries to control global warming & global warming is mainly due to emission of green house gasses i.e mainly due to carbon die oxide. Now worldwide regulation has been made to control carbon die oxide emission in production centre, transportation sector, logistic sector etc. If they emit Co 2 , that company has to paid tax for that. In future we can develop the models in such a way that, how firms manage carbon footprints in transportation problems under the carbon emission trading mechanism.

5)
In practical, the cargo transportation is happened in step by step system i.e., the real life transportation is happened from sources to destination centers via different intermediate destination centers. Taking this concept, in future, multi-stage four dimensional solid transportation problems can be formulated.